Surface cubications mod flips Louis Funar
18 pages
English

Surface cubications mod flips Louis Funar

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Surface cubications mod flips Louis Funar Institut Fourier BP 74, UMR 5582 CNRS, University of Grenoble I, 38402 Saint-Martin-d'Heres cedex, France e-mail: September 20, 2007 Abstract Let ? be a compact surface. We prove that the set of marked surface cubications modulo flips, up to isotopy, is in one-to-one correspondence with Z/2Z ?H1(?, ∂?; Z/2Z). MSC Classification(2000): 05 C 10, 57 R 70, 55 N 22, 57 M 35. Keywords and phrases: cubication, surface, critical point, cobordism, immersion. 1 Introduction and statements Cubical complexes and marked cubications. A cubical complex is a finite dimensional complex C consisting of Euclidean cubes, such that the intersection of two of its cubes is a finite union of cubes from C, once a cube is in C then all its faces belong to C and each point has a neighborhood intersecting only finitely many cubes of C. A cubication of a topological manifold is a cubical complex that is homeomorphic to the manifold. If the manifold is a PL manifold then one requires that the cubication be combinatorial and compatible with the PL structure. Our definition of cubication is slightly more general than the usual one, because we don't require that the intersection of two cubes consists of a single cube but only a finite union of cubes.

  • cubical complexes

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  • boundary

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Surface cubications mod ips
Louis Funar
Institut Fourier BP 74, UMR 5582 CNRS,
University of Grenoble I,
38402 Saint-Martin-d’Heres cedex, France
e-mail: funar@fourier.ujf-grenoble.fr
September 20, 2007
Abstract
Let be a compact surface. We prove that the set of marked surface cubications modulo ips, up to isotopy, is in one-to-one
correspondence with Z=2Z H ( ;@ ; Z=2Z).1
MSC Classi cation (2000): 05 C 10, 57 R 70, 55 N 22, 57 M 35.
Keywords and phrases: cubication, surface, critical point, cobordism, immersion.
1 Introduction and statements
Cubical complexes and marked cubications. A cubical complex is a nite dimensional complex C consisting of
Euclidean cubes, such that the intersection of two of its cubes is a nite union of cubes from C, once a cube is in
C then all its faces belong to C and each point has a neighborhood intersecting only nitely many cubes of C. A
cubication of a topological manifold is a cubical complex that is homeomorphic to the manifold. If the manifold is
a PL manifold then one requires that the cubication be combinatorial and compatible with the PL structure. Our
de nition of cubication is slightly more general than the usual one, because we don’t require that the intersection of
two cubes consists of a single cube but only a nite union of cubes.
The study of simplicial complexes and manifold triangulations lay at the core of combinatorial topology. Cubical
complexes and cubications might o er an alternative approach since, despite their similarities, they present some new
features.
nAny triangulated manifold admits a cubication, since we can decompose ann-dimensional simplex inton+1 cubes
nof dimension n. For k = 1 to n we adjoin, inductively, the barycenter of each k-simplex in and join it with the
barycenters of its faces. This way we obtain the 1-skeleton of a cubical complex, as shown in the gure below for
n = 3.
Thus, roughly speaking, working with simplicial complexes is equivalent to working with cubical complexes, from
topological viewpoint. We will show however that cubications might encode extra topological information.
1


























It is more convenient to work instead with marked cubications, that is cubications C of the manifold M, which are
endowed with a PL-homeomorphism jCj ! M (called the marking) of its subjacent space jCj. Two markings are
isotopic if one obtains one from the other by right composition with a PL-homeomorphism of M isotopic to identity.
This amounts to give the isotopy class of an embedding of the skeleton of C in M. Marked cubications underlying a
given cubication C are acted upon transitively by the mapping class group of M.
Bi-stellar moves. We will consider below PL manifolds, i.e. topological manifolds endowed with triangulations
(called combinatorial) for which the link of each vertex is PL homeomorphic to the boundary of the simplex. Recall
that two simplicial complexes are are PL homeomorphic if they admit combinatorially isomorphic subdivisions. There
exist topological manifolds which have several PL structures and it is still unknown whether all topological manifolds
have triangulations (i.e. whether they are to simplicial complexes), without requiring them to be
combinatorial.
It is not easy to decide whether two given triangulations de ne or not the same PL structure. One di cult y is
that one has to work with arbitrary subdivisions and there are in nitely many distinct combinatorial types of such.
In the early sixties one looked upon a more convenient set of transformations permitting to connect PL equivalent
triangulations of a given manifold. The simplest proposal was the so-called bi-stellar moves which are de ned for n-
0 0dimensional complexes, as follows: we exciseB and replace it byB , whereB andB are complementary balls that are
n+1unions of simplexes in the boundary @ of the standard (n + 1)-simplex. It is obvious that such transformations
do not change the PL homeomorphism type of the complex. Moreover, U.Pachner ([26, 27]) proved in 1990 that
conversely, any two PL triangulations of a PL manifold (i.e. the two triangulations de ne the same PL structure) can
be connected by a sequence of bi-stellar moves. One far reaching application of Pachner’s theorem was the construction
of the Turaev-Viro quantum invariants (see [31]) for 3-manifolds.
Habegger’s problem on cubical decompositions. It is natural to wonder whether a similar result holds for
cubical decompositions, as well. The cubical decompositions that we consider will be PL decompositions that de ne
the same PL structure of the manifold.
Speci cally , N.Habegger asked ([17], problem 5.13) the following:
Problem 1. Suppose that we have two PL cubications of the same PL manifold. Are they related by the following set
0 0of moves: exciseB and replace it byB , whereB andB are complementary balls (union ofn-cubes) in the boundary
of the standard (n + 1)-cube?
These moves have been called cubical or bubble moves in [10, 11], and (cubical) ips in [4]. Notice that the ips did
already appear in the mathematical polytope literature ([5, 37]).
The problem above was addressed in ([10, 11]), where we show that, in general, there are topological obstructions for
two cubications being ip equivalent.
Notice that acting by cubical ips one can create cubications where cubes have several faces in common or pairs of
faces of the same cube are identi ed. Thus we are forced to allow this greater degree of generality in our de nition of
cubical complexes.
Related work on cubications. In the meantime this and related problems have been approached by several
people working in computer science or combinatorics of polytopes (see [4, 7, 9, 20, 29]). Notice also that the 2-
2dimensional case of the sphere S was actually solved earlier by Thurston (see [30]). Observe that there are several
terms in the literature describing the same object. For instance the cubical decompositions of surfaces are also called
quadrangulations ([23, 24, 25]) or quad surface meshes, while 3-dimensional cubical complexes are called hex meshes
in the computer science papers (e.g. [4]). We used the term cubulation in [10, 11].
Remark that there is some related work that has been done by Nakamoto (see [23, 24, 25]) concerning the equivalence
of cubications of the same order by means of two transformations (that preserve the number of vertices): the diagonal
slide, in which one exchanges one diameter of a hexagon for another, and the diagonal rotation, in which the neighbors
of a vertex of degree 2 inside a quadrilateral are switched. In particular, it was proved that any two cubications of a
closed orientable surface can be transformed into each other, up to isotopy, by diagonal slides and diagonal rotations
2if they have the same (and su cien tly large) number of vertices and if their 1-skeleta de ne the same mod 2 homology
classes. Moreover, one can do this while preserving the simplicity of the cubication (i.e. not allowing double edges).
Immersions and cobordisms. Let M be a n-dimensional manifold. Consider the set of immersions f : F ! M
0 0withF a closed (n 1)-manifold. Impose on it the following equivalence relation: (F;f) is cobordant to (F ;f ) if there
0 0exist a cobordism X between F and F , that is, a compact n-manifold X with boundary F tF , and an immersion
0 :X !MI, transverse to the boundary, such that =ff0g and 0 =f f1g.jF jF
Once the manifold M is xed, the set N(M) of cobordism classes of codimension-one immersions in M is an abelian
group with the composition law given by disjoint union.
Cubications vs immersions’ conjecture. Our approach in [10] to the ip equivalence problem aimed at nding
a general solution in terms of some algebraic topological invariants. Speci cally , we stated (and proved half of) the
following conjecture:
nConjecture 1. The set of marked cubical decompositions of the closed manifold M modulo cubical ips is in bijection
nwith the elements of the cobordism group of codimension one immersions into M .
The solution of this conjecture would lead to a quite satisfactory answer to the problem of Habegger.
Notice that, when a cubical move is performed on the cubicationC endowed with a marking, there is a natural marking
induced for the ipp ed cubication. Thus it makes sense to consider the set of marked cubications mod ips.
We proved in [10] the existence of a surjective map between the two sets.
Digression on smooth vs PL category. There might be several possible interpretations for the conjecture above.
We can work, for instance, in the PL category and thus cubications, immersions and cobordisms are supposed PL.
Generally speaking there is little known about the cobordism group of PL immersions and their associated Thom
spaces, in comparison with the large literature on smooth immersions. However, in the speci c case of codimension-
one immersions we are able to compare the relevant bordism groups. If M is a compact n-dimensional manifold then
PLthe bordism groupN (M) of PL codimension-k immersions up to PL cobordisms is given in homotopy theoreticalk
terms by the formula:
PL 1 1N (M) = [M;
S MPL(k)]k
kwhere PL(k) is the semi-simplicial group of PL germs of maps on R , MPL(k) is a suitable Thom space associated
to it (see e.g. [36]),
denotes the loop space and S denotes the reduced suspension, while [X;Y ] denotes the set
of homotopy classes of maps X ! Y . This follows along the same lines as the results of Wells ([33]), where is
considered only the smooth case, by using instead of the classical Smale-Hirsch theory on smooth immersions the
Hae iger-P oenaru classi cation of combinatorial immersions from ([14]).
On the other hand, when M is smooth, the bordism group N (M) of codimension-k smooth immersions is given byk
the similar formula from ([33]):
1 1N (M) = [M;
S MO(k)]k
where MO(k) is the Thom space associated to the orthogonal groupO(k).
From the general results of Kuiper and Lashof ([18]) concerning the unstable homotopy type of PL(k) one obtains
that the natural inclusion mapO(1),!PL(1) induces a weak homotopy equivalenceMO(1)!MPL(1). This result
was improved later by Akiba, Scott and Morlet ([1, 21, 28]) to weak homotopy equivalences MO(k) ! MPL(k) for
all k 3.
This shows that codimension-one immersions of PL manifolds into a smooth manifold are PL cobordant to a smooth
immersion and also that the existence of a PL cobordism between two smooth immersions implies the existence of a
smooth cobordism. Consequently, when the manifold M is smooth, we can use either PL or smooth immersions and
bordisms, as the associated groups are naturally isomorphic.
Smooth cubications. However, in the DIFF category it is appropriate to consider only those cubications which are
smooth. Smooth cubications are de ned following Whitehead’s de nition of smooth triangulations from ([34]), but we
have to change it slightly in order to apply to the more general cubical complexes considered here.
3Let M be a smooth manifold and C a cubical complex. A map f : jCj ! M is called smooth if the restriction of
f to each cube of C is smooth. Moreover, f is non-degenerate if all these restrictions are of maximal rank. Finally
f : jCj ! M is a smooth cubication of M if f is a non-degenerate homeomorphism onto M. According to ([22],
Theorem 8.4) this de nition is equivalent to Whitehead’s one, when applied to simplicial complexes.
In small dimensions (e.g. when the dimension is at most 3) the PL and DIFF categories are equivalent. In particular,
we can assume from now on that we are working in the DIFF category and all objects are smooth, unless the opposite
is explicitly stated.
nComputations of the cobordism group of immersions. Finding the cobordism group N(M ) of (smooth)
ncodimension-one immersions into the n-manifold M was reduced to a homotopy problem by the results of [32, 33],
as explained above. However, these techniques seem awkward to apply when one is looking for e ectiv e results. The
ngroupN(S ) of codimension-one immersions in then-sphere, up to cobordism, is the then-th stable homotopy group
1 1of RP (since the Thom spaceMO(1) is homotopy equivalent to RP ) and it was computed by Liulevicius ([19]) for
n 9 as follows:
n 1 2 3 4 5 6 7 8 9
n 3 4N(S ) Z=2Z Z=2Z Z=8Z Z=2Z 0 Z=2Z Z=16Z Z=2Z (Z=2Z) (Z=2Z)
2It is known that, if M denotes a closed surface, then:
2 2 2N(M ) H (M ;Z=2Z)H (M ;Z=2Z)= 1 2
3Using geometric methods Benedetti and Silhol ([3]) and further Gini ([13]) proved that, if M is a 3-manifold, then
3 3 3 3N(M ) H (M ;Z=2Z)H (M ;Z=2Z)H (M ;Z=8Z)= 1 2 3
the right side groups being endowed with a twisted product. The result has been extended to higher dimensional
manifolds in [12].
Manifolds with boundary. Habegger’s problem from above makes sense also for PL cubications of manifolds
with boundary. The question is whether two cubications that induce the same cubication on the boundary are ip
equivalent.
Let M be a compact n-dimensional manifold with boundary @M. We will consider then the proper immersions
f : F ! M with F a compact (n 1)-manifold with boundary @F. This means that @M is transversal to f and
1f (@M \f(F)) =@F.
In order to de ne the cobordism equivalence for proper immersions we need to introduce more general immersions
and manifolds. The compactn-manifoldX is a manifold with corners if X is a PL manifold whose boundary@X has
0 0 0 0a splitting@X =F[@F [0; 1][F , where@F =@F , andF;F are manifolds with boundary. One says thatF[F
is the horizontal boundary @ X, @F [0; 1] is the vertical boundary @ X and their intersection @F f0; 1g, is theH V
corners set.
Observe now thatM[0; 1] is naturally a manifold with corners ifM has boundary, by using the splitting@(M[0; 1]) =
M f0g[@M [0; 1][M f1g. Let X be as above. One de nes then an immersion :X ! M [0; 1] to be an
immersion of manifolds with corners if
1. is proper and preserves the boundary type, by sending the horizontal (resp. vertical) part into the horizontal
(resp. vertical) boundary. Moreover, is transversal to the boundary.
2. The restriction to the vertical part : @F [0; 1] ! @M [0; 1] is a product i.e. it is of the form ( x;t) =
( ( x; 0);t).
40 0 0We say that the proper immersion (F;f) is cobordant to (F ;f ) if there exist a cobordismX betweenF and F (and
0thus @F =@F ) which is a manifold with corners and a proper immersion of manifolds with corners :X !M I,
0such that =f f0g and 0 =f f1g.jF jF
nThe set of cobordism classes of immersions of codimension one manifolds with boundary into a given manifold M
with prescribed boundary immersion can be computed using the methods of [12].
The main result. The aim of this paper is to solve the extension of the cubications vs immersions conjecture in the
case of compact surfaces, possibly with boundary.
Theorem 1.1. The set of marked cubications of the compact surface with prescribed boundary mod cubical ips is
in one to one correspondence with the elements of Z=2ZH ( ;@ ; Z=2Z).1
The proof of this theorem, although elementary, uses some methods from geometric topology and Morse theory.
Remark 1.1. One can identify a marked cubication with an embedding of a connected graph in the surface, whose
complementary is made of squares. The theorem says that any two graphs like that are related by a sequence of cubical
ips and an isotopy of the surface.
Acknowledgements. We are grateful to the referee for many valuable comments and suggestions leading to a better
presentation and the simpli cation of some proofs.
2 Outline of the Proof
Immersions associated to cubications. We associate to each marked cubicationC of the n-dimensional manifold
M a codimension-one generic immersion’ :N !M (the cubical complexN is also called the derivative complexC C C
nin [2]) of a manifold N having one dimension less than M. Here is the construction. Each cube is divided into 2C
equal cubes byn hyperplanes which we call sections. When gluing together cubes in a cubical complex the sections are
glued accordingly. Then the union of the hyperplane sections form the image ’ (N ) of a codimension-one genericC C
immersion. The cubulated manifoldN is constructed as follows: consider the disjoint union of a set of (n 1)-cubesC
which is in bijection with the set of all sections, then glue together two (n 1)-cubes if their corresponding sections
are adjacent in M. The immersion ’ is tautological: it sends a cube of N into the corresponding section. If theC C
cubication C is smooth then N has a smooth structure and the immersion ’ can be made smooth by means of aC C
small isotopy. This connection between cubications and immersions appeared independently in [2, 10] but this was
presumably known to specialists long time ago (see e.g. [30]).
Surface cubications and admissible immersions. The case of the surfaces is even simpler to understand. The
immersion ’ (N ) is obtained by drawing arcs connecting the opposite sides for each square of the cubication CC C
and N is a disjoint union of several circles. The immersions which arise from cubications are required to some mildC
restrictions. First the immersion is normal (or with normal crossings), since it has only transversal double points.
All immersions encountered below will be normal crossings immersions. Since we can travel from one square of C to
any other square of C by paths crossing the edges of C it follows that the image of the immersion ’ (N ) shouldC C
be connected. On the other hand, by cutting the surface along the arcs of ’ (N ) we get a number of polygonalC C
disks. An immersion having these two properties was called admissible in [10]. Further we have a converse for the
construction given above. If j is an admissible immersion of circles in the surface then j is ’ for some cubicationC
C of . The abstract complexC is the dual of the partition of into polygonal disks by means of the arcs ofj. Since
j can have at most double points it follows that C is made of squares.
Cubical ips on surfaces. There are four di eren t ips (and their inverses) on a surface, that we denoted by
b ;b ;b and b . They are pictured below.1 2 3 3;1
5bb 1 3
b
3 1,b2
In particular, we have the ips denoted by the same letters that act on immersions of curves on surfaces. These
transformations are local moves in the sense that they change just a small part of the immersion that lives in a disk,
leaving the immersion unchanged outside this disk. Speci cally , here are the ip actions.
bb1 3
b
3 1,b2
1The invariant of cubications. We associate to each proper immersion : L ! of a disjoint union of circles
1 1and intervals L , two independent invariants, as follows. The image (L ) is a union of curves on and it can
be viewed as a singular 1-cycle of . Notice that circles lay in the interior of while intervals are properly immersed
and thus their endpoints lay on the boundary. We set then
1j () = [(L )]2H ( ;@ ; Z=2Z)1 1
1Further we denote by j ()2 Z=2Z the number of double points of (L ) mod two, and eventually2
j () = (j ();j ())2H ( ;@ ; Z=2Z) Z=2Z 1 2 1
Further we are able to de ne the invariant associated to cubications by means of the formula:
j(C) =j (’ )2H ( ;@ ; Z=2Z) Z=2Z C 1
Remark that we don’t need to know thatj factors through the cobordism groupN( ) in order to de ne the invariant.
Observe also that the boundary cubication is the disjoint union of polygons corresponding to the boundary circles and
thus the numbers of edges determines completely their combinatorial type. The main theorem above is a consequence
of the following more precise statement:
Theorem 2.1. Two marked cubications C and C of the compact surface are ip equivalent if and only if j(C ) =0 1 0
j(C ) and their boundaries agree.1
Remark 2.1. In the case of closed orientable surfaces our result is a consequence of the Nakamoto-Ota theorem ([25]).
In fact, we will prove in section 5 that the diagonal transformations introduced by Nakamoto can be written as products
of cubical ips. However, their method could not be used to cover the case where the surface is non-orientable or has
boundary. Remark however that a weaker result holds true for diagonal transformations on arbitrary closed surfaces
(see [23, 24]), in which one replaced marked cubications up to isotopy by marked cubications up to homeomorphism.
Remark 2.2. Our methods are not combinatorial, as was the case of the sphere (see [30, 10, 4]), since one uses in an
essential manner the identi c ation of H ( ;@ ; Z=2Z) Z=2Z with N( ) , which is of topological nature. The main1
interest in developing the topological proof below is that one can give an unifying treatment of all surfaces and the
hope that these arguments might be generalized to higher dimensions. However, it would be interesting to nd a direct
combinatorial proof that provides an algorithm which gives explicitly a sequence of ips connecting two cubications.
Such an algorithm can be obtained in the closed orientable case by using the diagonal slides.
6Consider two cubicationsC andC having the same invariants. The rst step in the proof of theorem 2.1 is to return0 1
to the language of immersions and show that:
Proposition 2.1. If C and C have the same boundary and j(C ) =j(C ) then ’ and ’ are cobordant immer-0 1 0 1 C C0 1
sions.
The second step is to use the existence of a cobordism in order to produce ips and prove that:
Proposition 2.2. If ’ and ’ are admissible immersions which are cobordant then there exists a sequence of ips0 1
which connect them.
These two propositions will end the proof of the theorem 2.1.
3 Cobordant immersions are ip equivalent
3.1 Flips, saddle andX-transformations relating cobordant immersions
Connecting the immersions by means of maps with higher singularities. This section is devoted to the proof
of proposition 2.2. Consider thus a proper immersion ’ :F ! [0; 1] of a surface F which is a cobordism between
the immersions’ and ’ .0 1
The image ’(F) is an immersed surface having therefore a set of nitely many triples points that we denote by
S (’(F)). The set S (’(F)) of double points of ’ (at the target) form a one-dimensional manifold, whose closure3 2
contains the triple points. We have then a strati cation of ’(F) by manifolds
’(F) =R(’(F))[S (’(F))[S (’(F))2 3
where R(’(F)) is the set of non-singular (or regular) points.
Our aim is to analyze the critical points of the restriction of the height function to ’(F), by taking into account the
singularities of ’(F). In order to de ne critical points properly we need more terminology. Note that R(’(F)) and
S (’(F)) are subsets of ’(F) which might cause some troubles because critical points on the closure of a stratum2
might belong to another stratum.
Any point p 2 ’(F) has an open neighborhood that is di eomorphic to one coordinate plane, the union of two
3coordinate planes or the union of the three coordinate planes in R , depending on whetherp2R(’(F));p2S (’(F))2
or p 2 S (’(F)). The images of coordinate planes by this di eomorphism are called the leaves of ’(F) around p.3
Actually the leaves are well-de ned only in a small neighborhood. A pointp2’(F) will be called critical forh if p is
critical either for the restriction of h to some leaf containing p, or for hj , or else p2S (’(F)). Moreover, by3S2(’(F))
using a small perturbation of’ that is identity on the boundary, we can assume that the restriction ofh to the leaves
is also a Morse function.
A consequence of the Morse theory is the following. If the interval [t ;t ] does not contain any critical value forh then1 2
1 1there exists a di eomorphism of ft g into ft g that sends ’(F)\h (t ) on ’(F)\h (t ). Thus changes1 2 1 2
1in the topology of the slice ’(F)\h (t) arise only at critical t.
A critical point p will be said to be unstable if there are at least two leaves aroundp and the restriction of h to some
leaf has a critical point at p. The other critical points are called stable.
Lemma 3.1. One can perturb slightly ’ by using an arbitrary small isotopy that is identity on the boundary such that
h has only stable critical points.
Proof. Letp be an unstable critical point and be a leaf that is critical for the restriction ofh atp. This is equivalent
to the fact that the gradient of h (which is nowhere zero since h is regular) is orthogonal to the tangent plane at .
Since the immersion is normal crossings any other leaf around p should be transverse to and thus the restriction
of h to that leaf is non-critical at p.
Use now a small perturbation of the extra leaf aroundp by an isotopy that moves the intersection arc\ o p. The
new intersection point is not anymore critical for the restriction of h at .
7

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